Evolution Strategies — (μ,λ)-ES and (μ+λ)-ES
Evolution Strategies (ES) are among the oldest population-based optimizers, developed by Ingo Rechenberg and colleagues in 1960s Germany; the (μ+λ) variant came from Schwefel (and J. Reichenbacher) in the 1970s. They are the classical ancestors of cma-es. Source: andrey-dik‘s MQL5 implementation, with recombination letting offspring “inherit the characteristics of all parents.”
How it works — comma vs. plus
μ parents generate λ offspring via recombination + (self-adaptive) mutation; the two variants differ in who survives:
- (μ,λ)-ES (“comma”) — select the best μ from the λ offspring only; parents always die. New children “compete with their parents for a limited number of places, which can lead to premature convergence to a local optimum.”
- (μ+λ)-ES (“plus”) — select the best μ from parents ∪ offspring combined; good parents persist, giving “a broader exploration of the solution space.” A textbook exploration-vs-exploitation knob: elitist retention (plus) vs. forced turnover (comma).
Benchmark (context-relative)
Reported on the mature %-of-MAX rating (Hilly/Forest/Megacity) — the same scale as the founding four, so directly comparable:
| Variant | Score | % of MAX |
|---|---|---|
| (μ+λ)-ES | 6.496 | 72.18% |
| (μ,λ)-ES | 4.610 | 51.22% |
(μ+λ)-ES achieved “phenomenal results” as the current leader of the suite, “ahead of its nearest competitor SDSm by almost 10%” — above all four founding algorithms (best of those: BSA at 55.10%). A 1970s elitist strategy topping the modern field is a sharp no-free-lunch-theorem twist: the “plus” tweak (keep parents) outperforms decades of newer metaphors. The comma variant lands mid-pack, showing the survival rule alone swings rank by ~20 points.
Related
metaheuristic-optimization · exploration-vs-exploitation · population-optimization-benchmark · no-free-lunch-theorem · andrey-dik · cma-es · backtracking-search-algorithm